## Involve: A Journal of Mathematics

- Involve
- Volume 11, Number 3 (2018), 489-500.

### A tale of two circles: geometry of a class of quartic polynomials

Christopher Frayer and Landon Gauthier

#### Abstract

Let $\mathcal{P}$ be the family of complex-valued polynomials of the form $p\left(z\right)=\left(z-1\right)\left(z-{r}_{1}\right){\left(z-{r}_{2}\right)}^{2}$ with $\left|{r}_{1}\right|=\left|{r}_{2}\right|=1$. The Gauss–Lucas theorem guarantees that the critical points of $p\in \mathcal{P}$ will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, the unit disk contains two “desert” regions, the open disk $\left\{z\in \u2102:\left|z-\frac{3}{4}\right|<\frac{1}{4}\right\}$ and the interior of $2{x}^{4}-3{x}^{3}+x+4{x}^{2}{y}^{2}-3x{y}^{2}+2{y}^{4}=0$, in which critical points of $p$ cannot occur. Furthermore, each $c$ inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in $\mathcal{P}$.

#### Article information

**Source**

Involve, Volume 11, Number 3 (2018), 489-500.

**Dates**

Received: 21 February 2017

Revised: 5 June 2017

Accepted: 13 June 2017

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513775079

**Digital Object Identifier**

doi:10.2140/involve.2018.11.489

**Mathematical Reviews number (MathSciNet)**

MR3733970

**Zentralblatt MATH identifier**

06817033

**Keywords**

geometry of polynomials critical points Gauss–Lucas theorem

#### Citation

Frayer, Christopher; Gauthier, Landon. A tale of two circles: geometry of a class of quartic polynomials. Involve 11 (2018), no. 3, 489--500. doi:10.2140/involve.2018.11.489. https://projecteuclid.org/euclid.involve/1513775079